 Thank you very much for the invitation. It's a pleasure to be here. Can you see this? Can you hear this? They don't have a microphone. Okay, it's okay. They will tell you. Okay, so my title is Equivariant Decap Modules on rigid analytics spaces. So I will try to explain at least what cap means. It's not demodules, it's some variant of demodules. I'll start with some motivation. Bang! Sorry. So the motivation in this story comes from the desire to understand representation theory of periodic groups. So, well, some branch of it. Let me be more precise. So L will be a finite extension of Qp. K will be some complete and discreetly valued extension. For simplicity. I haven't had time to work out all the details yet when K is more general. So L will be the field of definition of the group and K will be the field of coefficients in which the representations are taken. G will be a connected reductive algebraic group and Raymond G will be some open. For most of my life, in fact all of my life, I have worked with compact periodic groups, but I am now understanding that the theory should extend quite naturally to the general case, but in my mind usually G will be compact. So the problem is to understand the following category of representations, admissible, locally analytic, and locally L analytic K representations of G. I will not give the definition of this. I will only say that it's the opposite of the category of co-admissible modules over the so-called locally analytic distribution algebra of G. So let me write that down. It has been defined by Neider and Teitelbaum about 13 years ago, so 14, 15 years ago. It is some kind of distribution algebra. It's the strong dual of the ring of K-valued locally analytic functions on the group G, and I think of it as some completion of the group ring of G. So I don't want to spend my time discussing the beginning of this theory. It's sufficient to just think of it as a completion of the group ring. Which are supposedly of compact support, or can you give an idea geographically, or does this... Yes, I think of compact support here. So you take the... You define it for compact groups first, and then you take the direct limit over all possible open compact separate groups instead of the general G. Okay, so for a compact group, is that functional or some local analytic? For a compact group, it's functionals on the locally analytic functions on the group. Yeah, which are continuous when you fix our uses of convergence of things like this. Okay, I think I understand. Hopefully you will see the net flavor of it in what follows. Whatever it is, it's a fresh air space over the ground field K, and it contains the abstract group ring of the group G inside it. Great, it cannot be... It's important for my personal psychological bandage, I'm more familiar with it. I don't think at this stage that you don't need it for theory works just fine. Like for CP, or... It can take it to be, I think, any complete extension of the group. Yeah, because in some old talks they mentioned something about spherically complex... Yes, I asked Peter why they did that. They said, I didn't get a sense of answer, so I don't think it's necessary. I got an impression from the old times that they had the technical difficulty in functional analysis or something. Which I'm not going to be doing at all, so... It's all algebra. The question is whether there is a difficulty or not, or maybe it's just trivial to get around it. This is the problem. So, I don't know because so far I haven't been working with complete discrete values, if that's okay, just to be safe. One, two, five, oh well. Right. So, now, the advantage of completing this group ring in this particular way is that it's big enough to contain the universal and helping algebra of the algebra. Of the group. So, you can just differentiate local analytic functions in the direction of the field and then evaluate the resulting derivative at one and that gives you a linear functional on the ring of k-valued local analytic functions. And that gives you an embedding of the Lie algebra into a distribution algebra which extends by the universal property to a ring of some. And now, my starting point, starting point of this work is the a terrible historian so I will try not to miss anyone out. The following famous theorem due to Berlinsen-Bernstein-Verlinskin-Cachevoir tells you how to think about modules over the envelope management. Let me remind you for every dominant regular weight of the maximal torus and this works over an algebraic closure of the ground field works over an algebraic closed field of characteristic zero. Lambda of H alpha is not in the set minus one zero, one, two, three, four, five for every H alpha. Ah, not. Yeah, yeah. So, overseas, of course, it's geometric but here I'm just making the minimal conditions needed to make the proof go through. Right, so representations of a semi-simple algebra with a fixed action of a center, assuming it's nice enough, are equivalent to twisted D modules on the flag variety which are quasi-cache-oriented modules. So you can, this opens the door to applications of methods from algebraic geometry to the study of representation theory of Lie groups and the algebras and in particular led to the first proof of the cache-oriented connectors. So the goal of this work is to prove and analog this local analytic distribution algebra the hope being that this embedding is nice enough to support such an extension. Well, clearly in order to do this, I will need to understand the analysis side of things a bit better. Right? What is this completion of the group ring and in particular what is this is purely algebraic, this is some kind of analytic guy so what's the relationship when you understand the completion better so I'll try to give my take on it. So the first question from the point of view of this presentation to ask I think is, well, what is the closure of U of G inside DGK, so this is some kind of fresh algebra what is the closure of embedding algebra? I'll tell you. So define standard monomials in some basis for the algebra over K and let me make a definition. So here's the first time where these caps come in. What is this cap? It's a set of I'm sorry, I don't know standard notation for the set of non-commutative monomials but this means just the commutative bound series ring strips of its commutative multiplication just thought of a set of formal power series with fixed ordering no x2, x1, x2 just on this power series commutative power series but on it you can define the multiplication correct and I don't have to define it all of it only on those power series which are convergent everywhere on the dual of the algebra. So this definition seems to be dependent on the choice of basis. I've chosen it doesn't you can choose, check quite easily it doesn't depend on the choice of basis it's an intrinsic ring extension of the enveloping algebra it can be defined as algebraic way with no mention of completions, apologies. To make symmetry Symmetric algebra with the algebra is not as an algebra but as a vector space I don't know between and so it comes Okay, yes, yes I would have to take SFG I don't understand exactly what you are doing because if let us say you taken a billionly algebra then you you must then the symmetric algebra is the enveloping algebra so it is commutative but you claim to be taking non commutative power series so for every monomial all the permutations are different monomials in your so it's too much you get something bigger it will not be I don't see how you map for example UGK into this because you can write everything in UGK once you fix an ordering of the basis you can write everything by Pankare Birkhoff with in the sum of monomials which are arranged according to this ordering so then you can use it to define them I don't know exactly how you it's exactly what I said you symmetrize things what? I can't explain I haven't thought about the symmetrization argument for a while I believe in the paper of Kohlhauser where it computes the centre of a distribution algebra he does use the argument you mentioned I think it does work although I haven't looked it for a while I just want to think of it as this set and I can give it a ring structure in a unique way such that the obvious map from algebraic thing into this thing is a ring homomorphism what is obvious map? X i goes to X i the linear map X i X j minus X j X i equals a bracket X i but the relations are not verified in the but I've forgotten the ring structure here this is now purely a set of a vector space no more, forget about the actual ring structure here it's just a set of found series I don't want to multiply them in the usual way I'm asserting that there is a ring structure on it no but if you use this it will be too loud perhaps you won't how series arranged where the X i X d are arranged in their order that is only monomials of the form X i A i of the X d A d were not in the other ordering because if you have all those things then the relations that you mentioned you see X i X 2 minus X 2 it's when you can discuss it I will explain it to you as yet I'm sorry for not making this a little so let's just define that to be the errands micro envelope and then with theorem essentially due to Schneider-Tatterbaum it says that the closure is this errands micro envelope and so the strategy is to first localize u g k cap localize in an imprecise sense in the sense of proving the Malgus theorem to the Beddonston-Bernstein theorem listed there and then extend to the distribution algebra so I want to explain how to do this at least so and I want to say why rigid penalty quantization so another invariant way of presenting this set of power series converging power series is as the ring of rigid analytic everywhere globally convergent functions on the dual of the algebra and this is only as a fresh air space yes rigid analytic and again I've forgotten here and here I've forgotten the ring structure I just want to say this is true as vector spaces now one of the main path we lost Beijing so we lost Beijing so we wait just few minutes to get Beijing again if you want a precise definition of the ring structure on this set I will give it later I will give it a size only of which gives 3 no the question is I thought it is not the right set not that the because when the algebra is commutative you should use commutative power series so you say it's it's a very it's a very no but I didn't understand what is the set it let us say I didn't understand what is the set but we can explain it in the end so okay so maybe I can play the ball yeah okay so the gym is again here okay good so both of you I want to take the middle ball this is probably stupid so observation the the way localization works involves an algebraic quantization and so this is intuition let me try to make it a little bit more precise I will write down 3 rings and 2 ring maps connected them which are non commutative which are filtered which are associated with graded are geometric so this guy quantizes just affine d space what is u lambda so I take the enveloping algebra and I'm mod out by an ideal which is that ideal generated by maximum ideal of the center of the enveloping algebra generated by things that form z minus lambda of z that thing quantizes the nil cone whatever d lambda is it also has an increasing filtration ring filtration filtration of sheaves of rings which is associated with graded is basically the ring of functions on the cotangent bundle of the flag variety so there is there is a filtration on d lambda when you take this associated with graded and you take sheaves spec you get a variety which turns out to be the cotangent bundle of the flag variety and there is a well named map connecting the cotangent bundle of the flag variety with the nil cone called the spring resolution the cotangent bundle of the flag variety is a resolution of singularities of the nil cone that correspond to the infinitesimal center character yeah lambda is in this context I just mean guru of this is that nothing else I don't want to make anything this is just loss it will be very nice if I could make it more precise but it tells me how to proceed in the pietic situation to localize this well I can do the same thing in the pietic setting because I still have the highest chandra center sitting inside the envelope and I can still mod out by maximum ideals and that what we need to do to define some kind of completion of d lambda will be a rigid analytic quantization of the corresponding object namely the rigid analytic variety underlying this algebraic variety cotangent bundle of the flag variety so this was all motivation now I can maybe write down some precise statements and maybe the motivation will help you to understand why I'm writing them now so now I will try not to say anything else about twists because they're just confusing and don't present any technical difficulties that I found yet so untwisted decap modules is what I want to talk about next and this section is joint work with Simon Wadsley definition net x v n smooth affinoid variety over the ground field a li lattice is a finally generated submodule curly L of the derivations of Lx finally generated over the subring of power bounded elements such that well it's a lattice it's a li algebra on power bounded elements and so a more correct way to say this is that it's a k circle comma o x circle li algebra or li algebra over the integral structure given any such li lattice and there are plenty of such things because whenever I have a basis for t of x over o of x say v1 up to vd maybe the li brackets of these guys don't lie inside the o x circle submodule generated by them and multiply them by a huge power of pi and then they will be but to any li algebraoid I can associate it's in helping in vessel melting algebra u of L okay? it's the smallest thing generated by so what is a li algebraoid? you need a commutative ring and you need a li algebra well a module over the commutative ring so that means that the commutative rings acts on the li algebra and the li algebra then acts on the commutative ring so in the basic examples t of x and o of x and so the melting algebra of t of x is going to give you d of x okay? and this works in great generality all you need is just a commutative ring and the li algebra and the module the universal algebra completely general defined first by Hertz in 1953 and then Reinhardt and then it just seems to be lost I know about Reinhardt which is a little bit Hertz maybe H, E, R, Z I don't know so you have u of L but then I can periodically complete it and then I can invert pi just like we do in rigid geometry so this will be some non commutative algebra over K and under these assumptions actually will be nepherian but unfortunately it depends on this choice of lattice integral structure so we get rid of that choice by taking the inverse limit over all things and this is just sort of elegant but you only need just one li lattice and then big pi param multiples of it will be co-final big pi param multiples of any other li lattice and so I could have just done inverse limit of u pi to the NL K hat and so geometrically this corresponds to stretching along big balls inside the cotangent bundle I'm trying to do that now I will state theorems just have to know an example so here is a li lattice well obviously you can take K circle take X del X and multiply by big power of pi and it needs to be non-negative here otherwise it's not the algebra and then if I do this construction I will get a Banach algebra non commutative which as a K vector space and as a Banach K Tate X module is isomorphic to the Tate algebra in two variables K X and a new variable that's what it looks like you say that it is ah it is isomorphic as a module to the commutative guy okay this is what you say so and D cap is just the intersection of these non commutative so it contains in particular the Tate algebra K Tate X it will contain del so it will contain the D of X as it were so the hybrid thing affinoid in the base direction and polynomial in the cotangent direction and more stuff on top and that's what it is and this is M and I should say of course this is just O of T star X S and so because of this this is morally at least in my mind any rigid analytic quantization of T star X so um we've thought quite a lot about this now and we're pretty sure this is the right definition in particular it globalizes nicely so this definition satisfies Tate theorem so take the standard site associated to a rigid analytic variety just open it open affinoid subsets um or G topology in the sense of BGR if you prefer and then D cap is a sheath now it goes a sheath what goes a sheath? we're vanishing higher check how much um okay but it looks a bit wild isn't it if I want to have some kind of D cap modules I need some control I need some finalist conditions on modules over D cap X and sheaths of D cap modules fortunately Schneider and TattleBam have already worked this out they have defined a nice category of modules called codenysver modules in particular which will give a definition of what codenysver meant at the start of a talk because they did prove that the distribution are of any compact the other group is Thrashe Stein so that's the definition I'm about to give when you say vanishing higher check homology do you refer to check to coverings of X by affinoids or to coverings of an affinoid by smaller affinoids of both cases both cases both cases ah okay more affinoid is also affinoid and the definition is it's a functor you can cover an affinoid by finding many smaller ones take the check complex and you can say that this is a sick league and you can also cover X itself so you can take the homology of X itself which is not affinoid X is affinoid X is still affinoid I'm not saying okay I was confused still thinking that I will say when it's not you are in that books no no no problem thank you and so suppose we have a tower of an aetherian Banach algebras with I'm going to be sloppy and say flat transition maps I think the definition only needs one side maybe on the right in all cases I know it will be true on both sides suppose you have such a situation then you can take the inverse limit and then the definition is that the inverse limit is an example of a fresh aetherian algebras okay so what well this gives you first of all it's kind of nicely independent in the choice of presentation it can happen that you have two different towers giving rise to the same inverse limit SK fresh algebras but as long as you have one you can do the same whenever you have such a fresh aetherian algebras where you can define a code missable module a module is code missable whenever you say think of geometrically it's some sheave defined on the whole cotangent bundle whenever you pull back to some bounded affinoid subset it's coherent coherent basically I'm very sorry is a binding generator a n module for all n completely algebraic only algebraic no just algebraic yes algebraic tends to be and so it's quite amazing that with this definition you get a nice a b b category proof's not deep so let curly c of a b category of code missable n modules is quite missable the problem is that fresh aetherian algebras are not netherian unfortunately so there are lots of examples of binary generator modules which are not binary presented next result is that our d cap rings are always fresh aetherian affinoids the context there is an ambiguity whether you mean the current or the current exist in this category or they are equal to the ones in the bigger one all are immodest in the setting every abstraction morphism is continuous kernel is closed it's very nice but we're actually going to be so with this in hand I can define the global version of d cap coherent modules it must code missable if well obviously I would want its sections on an affinoid to be code missable in the sense I've just defined but I do need an extra condition which is analogous to quasi coherent in the algebra geometric world I'm very sorry can I not go into the details of this symbol there is a very nice and very obvious thing you can do you can do a completed tensor product basically tensor product as fresh aetherian spaces but you can make sense of it in the situation you require that whenever I have an inclusion of affinoids V inside U then basically M of V is completely determined by M of U in a functorate way so here X is not an affinoid that's correct sorry yes at this point I might say now for now on X is general okay so you glue the previous definition yes is it true that M is the inverse limit of M tensor A A N yes it is it follows from the definition or you have to put it as an extra condition I don't remember the technical details but I think you have to do a bit of work for that you have to show that I am one of the impenetrable people you have to do something I'm sorry I'm using it as a black box I don't remember the question is whenever the tens of tens of products are finally generated the module is the inverse limit something like this yes it's apparently defined by here so in fact you confused me no so far I think let me take it back that I have extended X is still affinoid here okay I've just defined sheeps just a still affinoid here I do have here now extend follow my program so can I remove this now yes you can remove it so with X still affinoid CX will be the category cognizable sheeps WX is still affinoid here but now there sheeps it that is an equivalence of categories between modules so cognizable modules over DCAP of X and sheeps which are cognizable and tensed defined and so it follows that the sheeps category is also I think I do mean it to be as a false category of the category of sheeps and modules so now we extend no extend by shiftification DCAP and CX to every smooth rigidness space and I think well we've got three definitions of CX but I think the one I gave actually works in general you facet where's it gone here so if you take this definition X is no longer affinoid you impose these conditions it's a theorem that this category is the same as that you would have got if you were to have a sheaf which had property that there was some covering such that on each covering its restriction was lock of something cognizable so it kills theorem can I lift so I am variable with boards can I erase this here's our next theorem suppose I have X which is a closed smooth variety of Y smooth rigid I can have DCAP modules on X and I have DCAP modules on Y and I can ask what's the relationship and so it turns out that this is when Y having zero sections away from X are equivalent to DCAP modules on X so I have stated these as theorems because I have written them up and I am 99% confident they are true anything after this is not written yet I am still confident the next one is true so maybe they are not written yet so maybe I'll put quotes around them to be safe the next expected theorem is in Bensland localization suppose I have a linear form on H for values in K H being the Lie algebra of a split maximal torus in G as before so let's suppose it's split for simplicity but it's a Lie algebra theorem so I don't care I'm dominant in the way I've defined earlier in regular meaning that the stabilizer in the shifted action is trivial then so actually I have not defined D lambda cap but we have a definition which is valid so you can define U of curly L cap for any K comma O X Lie algebraed curly L not just a tangent model you can take a twisted twisted thing so everything now here makes sense Bian is a smooth virginality variety you can talk about code miscible sheaves of D cap modules which are defined locally and glued together on affinoids and the when you take global sections you will get modules over the Aaron's Michael envelope and I should have said I'm very sorry one of the easiest examples of a freshly standard algebra is the Aaron's Michael envelope the Aaron's Michael envelope UG cap that I defined as a set of power series you can define it as U of L cap where L is a finite dimension of the algebra you take the inverse limit of the U of finite rank things so this gives you a bridge between modules over Aaron's Michael notes and these sheaves so it's completely full though it's expected it's not written I believe it to be completely proof but it's not written it's not written I mean at the very least I would be very confident if I was to impose the same condition on the prime P as I had in my other paper on formal things because there we needed to use techniques for physical cabinet up to actually compute global sections and there they for technical reasons various human P's are very good prime and so I think we can just write down a proof using that but this thing should not depend on P at all it's just heresy there anyway so why I didn't care well you can for example construct modules from geometry construct constructive reducible modules in the following way of course this is a an idea as old as the original localization theorem but it's nice that it works here so I have maybe a point in the flank variety maybe I have just trivial extra space one dimensional on it I can demodule push forward it to a skyscraper sheet on the flank variety take low sections and this will give me some kind of completed by my module for the algebra for example but of course I don't have to take a point take any close variety maybe with some something interesting on it so Pn is the it's the rigid analytic variety associated to the flank variety of the algebraic group 2 so it's proper in the sense of rigid analytic geometry and why Pn is a closed rigid analytic so it's also proper non-backwatch so it's a global right it's not just because it depends on which topology okay I mean you use okay I'm using the analytic closed analysis okay it depends what you mean by closed embedding okay with any affinity I understand it then sorry okay okay I said book I've missed one module over the irons recommend I'm out of time and I haven't said anything about a coherent things what do you want me to do so just proceed I think you can add more than 5 minutes not no more than 5 minutes okay sure and then I won't give you definition and maybe I'll just sketch and connect it again so let's just for safety G to be compact as before X to be a smooth binoid and G acts on X continuously and differentially so that's G is open yes open compact so he obviously by punctuality reserve the ring of paramount elements in O of X and induce an action on the reduction of X and let's assume that the action of G on the reduction has finite image and indeed the action of G on any O X circle one part of N O X circle is has finite image at least that so this is what happens when G is an open compact type of G of L acting on the flag variety you might have a small group stabilizing a smaller binoid in the flag variety then you can do the following construction it's you can take D cap of X D X cap is the same thing I can take the abstracts key group ring that's stupid it doesn't take into account topology of G and then I complete it a bit how to do that I look at various things involved in the definition of D cap there with these banner algebras and then if for any fixed L if I let G approach to 1 if I look at very small subgroups of G they will stabilize it in fact they will be acting as exponential of the algebra elements since I don't have time I'll just say that this is the inverse limit of all possible pairs L and N where this makes sense and alpha is just the derivative of the action of G let's say the algebra of G into T of X in this case because it's the algebra of G by the Baker-Campbell House of Formula this expression here is a group from N into the group of units so this makes sense and if I can define look at just G stable so it might be empty if G is big enough so that's not good but if G is small it will have at least some elements in it and then expected theorem 6 X W slash G just notation I want to use that notation to define that set this thing should be achieved on this G topology we'll get the finite coverings again a freshly done algebra and again vanishing higher and finally let me just remark that there's another definition let X mod G sub W be G stable and polycompact capons finite unions of things which are G stable then this is a good G topology because while sheaves on the rigid space are the same things as sheaves on the attic space in the sense of Hubert and G acts on the attic space so I can take X attic mod G as topological spaces I give it the quotient topology and then I pull back the site structure X at the moment is just a phenoid classical rigid space sorry none of this is work in progress all this is work in progress let me write down a remark on the theorem so that's a Hubert space and so the expected theorem is well if you believe theorem 6 then you might believe that I can define some version of codemissible d cap star G modules in the same way gluing locally at least fasciae stone finally 7 then lambdas before then I can write down codemissible of course this has not been defined in this talk but whatever it is it's going to be equivalent to what I want mainly the objects on the representation of the attic side which are codemissible modules of over distribution algebra which are dual of admissible representations which occur in numbers and of course because I'm working with lambdas I need to improve to fix anything decimal and sorry for taking so long thank you very much so maybe we take first questions from Tokyo where is the if there is no questions from Tokyo then we go to BG any questions in BG so I have a question so in your talk how do you define the category CX for the smooth rich analytic space and suppose you have a morphism for example pro-morphism of smooth rich analytic space X to Y then how do you take the category CX and CY does there exist any function from CX to CY in the case of closed embedding we have a well defined which features in the statement for I plus I believe that it will also work in the proper case but we have not checked the details there will be problems in the case one the morphism is not and problem if you answer it for closed embedding it is okay and for other for morphism you have not checked yet that's right but it should be okay it should be okay okay thank you other questions from BG no I take a note so you can take few questions from us there was a question in the beginning yes so there was a question of foundations in the beginning what is the definition of those formal series yes we have to discuss them we postpone them just after we can but are there other questions do you have other questions? no because this is very far from what I can because I'm not really I don't know precisely so I cannot ask questions so maybe I have just one question so if you take the point of your formal scheme yes I forgot to apologize that my abstract said via aquarium formal models that's because I came up with this definition the last month beforehand I didn't know how to do things properly so I was going to go via your book and you buy formal models and you take the geographic objects and they may not be perfect so you can form but it's just to go on the main project thank you so I think that's all let's thank the speaker again